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G = C4.Q89C4order 128 = 27

7th semidirect product of C4.Q8 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C4.Q89C4, C4⋊C4.12Q8, C4.24(C4×Q8), C2.12(C4×SD16), C2.4(D4.Q8), C2.3(Q8⋊Q8), (C2×C4).111SD16, C23.777(C2×D4), (C22×C4).692D4, C22.157(C4×D4), C22.62(C4○D8), C22.4Q16.8C2, (C22×C8).43C22, C4.12(C42.C2), C22.62(C2×SD16), C4.14(C422C2), C4.23(C42⋊C2), C22.81(C8⋊C22), (C2×C42).292C22, C22.76(C22⋊Q8), C2.12(SD16⋊C4), (C22×C4).1375C23, C2.4(C23.46D4), C2.5(C23.20D4), C22.71(C8.C22), C23.65C23.5C2, C22.7C42.32C2, C22.89(C22.D4), C2.10(C23.63C23), (C4×C4⋊C4).15C2, C4⋊C4.77(C2×C4), (C2×C8).110(C2×C4), (C2×C4).270(C2×Q8), (C2×C4.Q8).16C2, (C2×C4).572(C4○D4), (C2×C4⋊C4).770C22, (C2×C4).393(C22×C4), SmallGroup(128,651)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.Q89C4
C1C2C22C2×C4C22×C4C2×C4⋊C4C4×C4⋊C4 — C4.Q89C4
C1C2C2×C4 — C4.Q89C4
C1C23C2×C42 — C4.Q89C4
C1C2C2C22×C4 — C4.Q89C4

Generators and relations for C4.Q89C4
 G = < a,b,c,d | a4=d4=1, b4=a2, c2=a-1b2, ab=ba, cac-1=a-1, ad=da, cbc-1=b3, dbd-1=ab3, cd=dc >

Subgroups: 220 in 115 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×10], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×20], C23, C42 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4.Q8 [×4], C2×C42, C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4, C22×C8 [×2], C22.7C42, C22.4Q16 [×3], C4×C4⋊C4, C23.65C23, C2×C4.Q8, C4.Q89C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, SD16 [×2], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C2×SD16, C4○D8, C8⋊C22, C8.C22, C23.63C23, C4×SD16, SD16⋊C4, Q8⋊Q8, D4.Q8, C23.46D4, C23.20D4, C4.Q89C4

Smallest permutation representation of C4.Q89C4
Regular action on 128 points
Generators in S128
(1 60 5 64)(2 61 6 57)(3 62 7 58)(4 63 8 59)(9 111 13 107)(10 112 14 108)(11 105 15 109)(12 106 16 110)(17 104 21 100)(18 97 22 101)(19 98 23 102)(20 99 24 103)(25 96 29 92)(26 89 30 93)(27 90 31 94)(28 91 32 95)(33 73 37 77)(34 74 38 78)(35 75 39 79)(36 76 40 80)(41 81 45 85)(42 82 46 86)(43 83 47 87)(44 84 48 88)(49 66 53 70)(50 67 54 71)(51 68 55 72)(52 69 56 65)(113 121 117 125)(114 122 118 126)(115 123 119 127)(116 124 120 128)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)
(1 107 58 15)(2 110 59 10)(3 105 60 13)(4 108 61 16)(5 111 62 11)(6 106 63 14)(7 109 64 9)(8 112 57 12)(17 87 102 41)(18 82 103 44)(19 85 104 47)(20 88 97 42)(21 83 98 45)(22 86 99 48)(23 81 100 43)(24 84 101 46)(25 79 94 33)(26 74 95 36)(27 77 96 39)(28 80 89 34)(29 75 90 37)(30 78 91 40)(31 73 92 35)(32 76 93 38)(49 113 72 127)(50 116 65 122)(51 119 66 125)(52 114 67 128)(53 117 68 123)(54 120 69 126)(55 115 70 121)(56 118 71 124)
(1 78 127 103)(2 37 128 23)(3 80 121 97)(4 39 122 17)(5 74 123 99)(6 33 124 19)(7 76 125 101)(8 35 126 21)(9 32 66 84)(10 90 67 43)(11 26 68 86)(12 92 69 45)(13 28 70 88)(14 94 71 47)(15 30 72 82)(16 96 65 41)(18 58 40 113)(20 60 34 115)(22 62 36 117)(24 64 38 119)(25 56 85 106)(27 50 87 108)(29 52 81 110)(31 54 83 112)(42 105 89 55)(44 107 91 49)(46 109 93 51)(48 111 95 53)(57 73 120 98)(59 75 114 100)(61 77 116 102)(63 79 118 104)

G:=sub<Sym(128)| (1,60,5,64)(2,61,6,57)(3,62,7,58)(4,63,8,59)(9,111,13,107)(10,112,14,108)(11,105,15,109)(12,106,16,110)(17,104,21,100)(18,97,22,101)(19,98,23,102)(20,99,24,103)(25,96,29,92)(26,89,30,93)(27,90,31,94)(28,91,32,95)(33,73,37,77)(34,74,38,78)(35,75,39,79)(36,76,40,80)(41,81,45,85)(42,82,46,86)(43,83,47,87)(44,84,48,88)(49,66,53,70)(50,67,54,71)(51,68,55,72)(52,69,56,65)(113,121,117,125)(114,122,118,126)(115,123,119,127)(116,124,120,128), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,107,58,15)(2,110,59,10)(3,105,60,13)(4,108,61,16)(5,111,62,11)(6,106,63,14)(7,109,64,9)(8,112,57,12)(17,87,102,41)(18,82,103,44)(19,85,104,47)(20,88,97,42)(21,83,98,45)(22,86,99,48)(23,81,100,43)(24,84,101,46)(25,79,94,33)(26,74,95,36)(27,77,96,39)(28,80,89,34)(29,75,90,37)(30,78,91,40)(31,73,92,35)(32,76,93,38)(49,113,72,127)(50,116,65,122)(51,119,66,125)(52,114,67,128)(53,117,68,123)(54,120,69,126)(55,115,70,121)(56,118,71,124), (1,78,127,103)(2,37,128,23)(3,80,121,97)(4,39,122,17)(5,74,123,99)(6,33,124,19)(7,76,125,101)(8,35,126,21)(9,32,66,84)(10,90,67,43)(11,26,68,86)(12,92,69,45)(13,28,70,88)(14,94,71,47)(15,30,72,82)(16,96,65,41)(18,58,40,113)(20,60,34,115)(22,62,36,117)(24,64,38,119)(25,56,85,106)(27,50,87,108)(29,52,81,110)(31,54,83,112)(42,105,89,55)(44,107,91,49)(46,109,93,51)(48,111,95,53)(57,73,120,98)(59,75,114,100)(61,77,116,102)(63,79,118,104)>;

G:=Group( (1,60,5,64)(2,61,6,57)(3,62,7,58)(4,63,8,59)(9,111,13,107)(10,112,14,108)(11,105,15,109)(12,106,16,110)(17,104,21,100)(18,97,22,101)(19,98,23,102)(20,99,24,103)(25,96,29,92)(26,89,30,93)(27,90,31,94)(28,91,32,95)(33,73,37,77)(34,74,38,78)(35,75,39,79)(36,76,40,80)(41,81,45,85)(42,82,46,86)(43,83,47,87)(44,84,48,88)(49,66,53,70)(50,67,54,71)(51,68,55,72)(52,69,56,65)(113,121,117,125)(114,122,118,126)(115,123,119,127)(116,124,120,128), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,107,58,15)(2,110,59,10)(3,105,60,13)(4,108,61,16)(5,111,62,11)(6,106,63,14)(7,109,64,9)(8,112,57,12)(17,87,102,41)(18,82,103,44)(19,85,104,47)(20,88,97,42)(21,83,98,45)(22,86,99,48)(23,81,100,43)(24,84,101,46)(25,79,94,33)(26,74,95,36)(27,77,96,39)(28,80,89,34)(29,75,90,37)(30,78,91,40)(31,73,92,35)(32,76,93,38)(49,113,72,127)(50,116,65,122)(51,119,66,125)(52,114,67,128)(53,117,68,123)(54,120,69,126)(55,115,70,121)(56,118,71,124), (1,78,127,103)(2,37,128,23)(3,80,121,97)(4,39,122,17)(5,74,123,99)(6,33,124,19)(7,76,125,101)(8,35,126,21)(9,32,66,84)(10,90,67,43)(11,26,68,86)(12,92,69,45)(13,28,70,88)(14,94,71,47)(15,30,72,82)(16,96,65,41)(18,58,40,113)(20,60,34,115)(22,62,36,117)(24,64,38,119)(25,56,85,106)(27,50,87,108)(29,52,81,110)(31,54,83,112)(42,105,89,55)(44,107,91,49)(46,109,93,51)(48,111,95,53)(57,73,120,98)(59,75,114,100)(61,77,116,102)(63,79,118,104) );

G=PermutationGroup([(1,60,5,64),(2,61,6,57),(3,62,7,58),(4,63,8,59),(9,111,13,107),(10,112,14,108),(11,105,15,109),(12,106,16,110),(17,104,21,100),(18,97,22,101),(19,98,23,102),(20,99,24,103),(25,96,29,92),(26,89,30,93),(27,90,31,94),(28,91,32,95),(33,73,37,77),(34,74,38,78),(35,75,39,79),(36,76,40,80),(41,81,45,85),(42,82,46,86),(43,83,47,87),(44,84,48,88),(49,66,53,70),(50,67,54,71),(51,68,55,72),(52,69,56,65),(113,121,117,125),(114,122,118,126),(115,123,119,127),(116,124,120,128)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)], [(1,107,58,15),(2,110,59,10),(3,105,60,13),(4,108,61,16),(5,111,62,11),(6,106,63,14),(7,109,64,9),(8,112,57,12),(17,87,102,41),(18,82,103,44),(19,85,104,47),(20,88,97,42),(21,83,98,45),(22,86,99,48),(23,81,100,43),(24,84,101,46),(25,79,94,33),(26,74,95,36),(27,77,96,39),(28,80,89,34),(29,75,90,37),(30,78,91,40),(31,73,92,35),(32,76,93,38),(49,113,72,127),(50,116,65,122),(51,119,66,125),(52,114,67,128),(53,117,68,123),(54,120,69,126),(55,115,70,121),(56,118,71,124)], [(1,78,127,103),(2,37,128,23),(3,80,121,97),(4,39,122,17),(5,74,123,99),(6,33,124,19),(7,76,125,101),(8,35,126,21),(9,32,66,84),(10,90,67,43),(11,26,68,86),(12,92,69,45),(13,28,70,88),(14,94,71,47),(15,30,72,82),(16,96,65,41),(18,58,40,113),(20,60,34,115),(22,62,36,117),(24,64,38,119),(25,56,85,106),(27,50,87,108),(29,52,81,110),(31,54,83,112),(42,105,89,55),(44,107,91,49),(46,109,93,51),(48,111,95,53),(57,73,120,98),(59,75,114,100),(61,77,116,102),(63,79,118,104)])

38 conjugacy classes

class 1 2A···2G4A···4H4I···4R4S4T4U4V8A···8H
order12···24···44···444448···8
size11···12···24···488884···4

38 irreducible representations

dim11111112222244
type++++++-++-
imageC1C2C2C2C2C2C4Q8D4SD16C4○D4C4○D8C8⋊C22C8.C22
kernelC4.Q89C4C22.7C42C22.4Q16C4×C4⋊C4C23.65C23C2×C4.Q8C4.Q8C4⋊C4C22×C4C2×C4C2×C4C22C22C22
# reps11311182248411

Matrix representation of C4.Q89C4 in GL6(𝔽17)

0160000
100000
0016200
0016100
0000160
0000016
,
5120000
550000
000700
005700
000001
000010
,
1010000
170000
0013800
000400
000040
000004
,
1300000
0130000
001000
000100
000001
0000160

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,5,0,0,0,0,7,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[10,1,0,0,0,0,1,7,0,0,0,0,0,0,13,0,0,0,0,0,8,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0] >;

C4.Q89C4 in GAP, Magma, Sage, TeX

C_4.Q_8\rtimes_9C_4
% in TeX

G:=Group("C4.Q8:9C4");
// GroupNames label

G:=SmallGroup(128,651);
// by ID

G=gap.SmallGroup(128,651);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,394,2804,718,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=d^4=1,b^4=a^2,c^2=a^-1*b^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^3,d*b*d^-1=a*b^3,c*d=d*c>;
// generators/relations

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